Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-5x-2y &= 4 \\ -3x+2y &= -8\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Add the top and bottom equations. $-8x = -4$ Divide both sides by $-8$ and reduce as necessary. $x = \dfrac{1}{2}$ Substitute $\dfrac{1}{2}$ for $x$ in the top equation. $-5( \dfrac{1}{2})-2y = 4$ $-\dfrac{5}{2}-2y = 4$ $-2y = \dfrac{13}{2}$ $y = -\dfrac{13}{4}$ The solution is $\enspace x = \dfrac{1}{2}, \enspace y = -\dfrac{13}{4}$.